Metastable Distributions of Semi-Markov Processes

2024/11/07 by Koralov, Leonid, Imtiyas, Ishfaaq Mohammed · 1 citation
#60J27 #60K15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2411.04795

Abstract

In this paper, we consider semi-Markov processes whose transition times and transition probabilities depend on a small parameter ε. Understanding the asymptotic behavior of such processes is needed in order to study the asymptotics of various randomly perturbed dynamical and stochastic systems. The long-time behavior of a semi-Markov process X^ε_t depends on how the point (1/ε, t(ε)) approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale t(ε). The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.

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